Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{z^2 - 4}{z + 2}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{4} = 2$ So we can rewrite the expression as: $r = \dfrac{({z} + {2})({z} {-2})} {z + 2} $ We can divide the numerator and denominator by $(z + 2)$ on condition that $z \neq -2$ Therefore $r = z - 2; z \neq -2$